Analogue and digital signals can carry the same information under certain conditions. Answered by
Harry Nyquist and Claude Shannon. Under appropriate "slowness" conditions for x(t) we
have the Sampling Theorem:
x(t) = \sum_{n=-\infty}^{\infty} x[n] \mathrm{sinc}\left(\frac{t - nT_s}{T_s}\right)
We can build the continuous time signal from the discrete time sequence. Take copies of the \mathrm{sinc}()
function at each sample location scaled by the amplitude of the sample and sum them to get back the
original function.
The conditions under which you can do this are determined by the Fourier transform. Once we know
the "speed" of the signal we can choose a sampling interval that will allow the above
theorem to work - this is the Nyquist rate.

Discrete time signal is a sequence of complex numbers denoted x[n] where the square
brackets are used to indicate its discrete nature. The index n just provides an ordering for
samples which are taken at a steady interval, the sampling period. The sequence is two-sided in
that it goes from minus infinity to plus infinity.

infinite length - index N ranges over entire range of integers. abstract. good for theorems. They have infinite energy.

periodic - data repeats every N samples. Represent with a tild on top. Same info as a finite-length of length N. They have infinite energy.

finite support - infinite length with on a finite number of non zero samples. Eg unit step.

Elementary operations include scaling, sum, product. These can be applied to any discrete signal.
Shift-by-k can be applied to infinite signals, but when applying to discrete time signals we need
to state what happens when we go beyond the index range N, i.e., how we embed it into an
infinite length sequence. Can embed into a finite-support sequence by putting zeros on the
left and right. Or we could use a periodic extension, whereby the shift becomes circular.
The periodic extension is the natural way to interpret the shift of a finite-length signal.